Elastic lattices for design of tensegrity structures and robots

ABSTRACT

According to some embodiments of the invention, a tensegrity robot includes a plurality of compressive members; and a plurality of interconnecting tensile members connected to the plurality of compressive members to form a spatially defined structure without the plurality of compressive members forming direct load-transmitting connections with each other. The plurality of interconnecting tensile members forms a lattice, and the lattice comprises an elastic material.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to U.S. Provisional Application No.62/466,913 filed Mar. 3, 2017, the entire contents of which are herebyincorporated by reference.

STATEMENT OF GOVERNMENT SUPPORT

This invention was made with Government support under grant NNX15AD74G,awarded by the National Aeronautics and Space Administration (NASA). TheGovernment has certain rights in the invention.

BACKGROUND 1. Technical Field

Embodiments of this invention relate to robots, and more particularly totensegrity robots and methods of producing tensegrity robots.

2. Discussion of Related Art

Much prior work from the authors has demonstrated that tensile-integritysystems, called “tensegrity” systems, are useful for a wide range ofapplications, including space exploration in various ways [1-2, 5-8].Spherical tensegrity structures can be actuated to roll along unevenground [1-2, 5-7] and spine-like tensegrity structures can act as partof a larger robot that can walk [8]. These robots capitalize on thebeneficial properties of tensegrity systems, such as low mass, variablestiffness, and redundancy to failure.

However, designing and prototyping a tensegrity robot is atime-intensive task. The robots' cables require compliance forlocomotion, which is usually achieved by a combination of stiff braidedcable and springs [5-8] or by elastic cord [2]. The designer generallyhas requirements for the system's compliance; commonly, a symmetrictension network is required. The nature of the distribution of forces inthe elastic cables of the structure's tension network, as well as theprecision required in adjusting the length of the cables to achieve thedesired properties of compliance, create challenges for assembly andrepeatability.

Assembly using either of the prior cable methods requires repetitiveattachment, adjustment, and reattachment of the cables (generally doneby tying and untying knots) to achieve symmetric cable lengths. Theassembly process takes on the order of hours [6] or even weeks [7] tomake a completed tensegrity, and it is prone to error in achievingsymmetry. Using either of the prior cable methods also hindersrepeatability, since the cables are manually adjusted for each robot andthe resultant compliance is imprecise. In order to better design andevaluate different tensegrity systems, a more time-efficient, robust,and repeatable method is needed for tensegrity prototyping.

SUMMARY

According to some embodiments of the invention, a tensegrity robotincludes a plurality of compressive members; and a plurality ofinterconnecting tensile members connected to the plurality ofcompressive members to form a spatially defined structure without theplurality of compressive members forming direct load-transmittingconnections with each other. The plurality of interconnecting tensilemembers forms a lattice, and the lattice comprises an elastic material.

According to some embodiments of the invention, a tensegrity robotincludes a plurality of compressive members; a plurality of firstinterconnecting tensile members connected to the plurality ofcompressive members to form a spatially defined structure without theplurality of compressive members forming direct load-transmittingconnections with each other; a plurality of second tensile membersconnected to the plurality of compressive members, each of the pluralityof second tensile members being in parallel to one of the plurality offirst interconnecting tensile members; a plurality of actuators, eachattached to one of the plurality of compressive members; and acontroller in communication with the plurality of actuators. Theplurality of first interconnecting tensile members forms a lattice, andthe lattice comprises an elastic material. Each actuator of theplurality of actuators is operatively connected to a corresponding oneof the plurality of second tensile members so as to selectively change atension on the corresponding one of the plurality of second tensilemembers in response to commands from the controller to thereby change acenter of mass of the tensegrity robot to effect movement thereof.

According to some embodiments of the invention, a method of forming atensegrity robot includes cutting a plurality of interconnecting tensilemembers from a sheet of elastic material; and connecting the pluralityof interconnecting tensile members to a plurality of compressive membersto form a spatially defined structure without the plurality ofcompressive members forming direct load-transmitting connections witheach other. The plurality of interconnecting tensile members forms alattice.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a tensegrity robot according to some embodiments of theinvention.

FIG. 2A shows passive tensegrity robot according to some embodiments ofthe invention.

FIG. 2B show actuated tensegrity robots according to some embodiments ofthe invention.

FIG. 3 shows the TT-4_(mini) robot, a tensegrity robot that uses theelastic lattice platform. This robot moves by adjusting the lengths ofits cables with respect to its elastic lattice.

FIG. 4 shows the tensegrity robot TT-3. All tensile members includecables and springs, each of which was individually tied together.

FIG. 5 shows a modular elastic lattice according to some embodiments ofthe invention, made with 60 A durometer rubber.

FIG. 6 shows a single-piece elastic lattice for a 6-bar tensegritystructure according to some embodiments of the invention.

FIG. 7 shows a step-by-step assembly sequence for a 6-bar tensegritystructure according to some embodiments of the invention.

FIG. 8 shows three 12-bar tensegrity structures constructed using anelastic lattice. The structures are robust and quick to assemble, andthey allow significant control over the systems' properties.

FIGS. 9A and 9B show two spine tensegrity structures constructed usingprototyping methods described herein. FIG. 9B shows the horizontal vs.saddle connectors: There are four horizontal strips of lattice material,compared to the many almost-vertical saddle connectors.

FIG. 10 shows a step-by-step assembly sequence for forming a tensegrityspine structure according to some embodiments of the invention.

FIG. 11 shows the TT-4_(mini) prototype performing punctuated uphillrolling on an inclined surface of 24°. The photo shows three steps bythe robot.

FIG. 12 shows visualization of the single-cable actuation policy. Eachrow corresponds to one cable, this policy can be repeated indefinitely.

FIG. 13 shows simulation results of the TT-4_(mini's) payload CoMtrajectory while climbing a 16° incline using the single-cable actuationpolicy.

FIG. 14 shows required percent of cable retraction to initiate forwardrolling motion with single-cable policy. Increasingly negativepercentages signify greater cable retraction.

FIG. 15 shows visualizations of the two-cable actuation policies. Eachrow corresponds to one cable, and each policy can be repeatedindefinitely.

FIG. 16 shows a summary of Hardware Experiment results.

FIG. 17 shows simulation results of the TT-4_(mini)'s payload CoMtrajectory while climbing a 26° incline using the alternating two-cableactuation policy.

FIG. 18 shows comparison of robot CoM height over time as a percent ofneutral stance CoM height for single-cable and two-cable actuationpolicies. Maximum heights for each policy shown as dotted lines.

FIG. 19 Shown in both simulation and hardware, the payload's CoM heightat the robot's neutral state for single-cable actuation (left) is higherthan that of the multi-cable actuation policy (right). The base polygonis highlighted in the lower figure.

FIG. 20 shows comparison of projected CoM (circles) with supporting basepolygons for single-cable (red) and two-cable actuation (blue) policieson a 10° incline. Direction of uphill travel is along the positive xaxis. Distance from the uphill edge of the robot's base polygon (dottedlines) is less for multi-cable actuation than single-cable actuation.Similarly, distance from the downhill edge of the robot's base(dash-dotted lines) is greater for the multi-cable policy than thesingle-cable policy.

FIG. 21 shows the TT-4_(mini) prototype climbing up a 24° inclinesurface with two-cable alternating actuation.

DETAILED DESCRIPTION

Some embodiments of the current invention are discussed in detail below.In describing embodiments, specific terminology is employed for the sakeof clarity. However, the invention is not intended to be limited to thespecific terminology so selected. A person skilled in the relevant artwill recognize that other equivalent components can be employed andother methods developed without departing from the broad concepts of thecurrent invention. All references cited anywhere in this specification,including the Background and Detailed Description sections, areincorporated by reference as if each had been individually incorporated.

According to some embodiments of the invention, a platform forprototyping tensegrity robots and structures is provided thatsignificantly reduces the time required for manufacturing and assembly,as well as increases precision and repeatability of the tensioned robot.This platform simplifies tensegrity system design and allows for morescientific experiments to be performed in less time. According to someembodiments, the new platform uses a rapidly manufactured modularelastic lattice as the tensile members in a tensegrity structures. Theelastic lattice can be laser-cut out of a sheet of elastic material,then wrapped around the bars or other of a tensegrity structure,replacing the traditional cables and springs that are more commonlyused. Production of the elastic lattice is efficient, as laser cuttingis straightforward and fast. Wrapping the elastic lattice around therods of a tensegrity system takes on the order of minutes, rather thanon the order of hours that it takes to assemble a tensegrity usingtraditional methods. This prototyping platform has been used to createboth type I and type II tensegrity systems using lattice designs eithercut out of a single sheet or with individual smaller lattices. Inparticular, this prototyping platform has been used to make 6-barspherical, 12-bar spherical, and spine-like tensegrity structures.

Embodiments of the invention address the challenges of rapidlyprototyping and manufacturing tensegrity structures. According to someembodiments, a machine-cut lattice of elastic material is used in placeof the spring/cable system in a tensegrity structure, automating themanufacturing of the tension elements and greatly simplifying theassembly processes, as well as creating consistent and repeatableprototypes. According to some embodiments, a design is made in 2D usingcomputer-aided design software that is used to automatically laser-cutan elastic sheet. After the rigid elements of the system (rods, or other3D structures) are manufactured separately, the laser-cut design can beattached to the rigid elements in a particular pattern, creating thetensegrity structure. This process is fast and efficient. Additionally,because the tensile members (the elastic lattice) are manufactured by amachine, e.g., a laser cutter, instead of created by hand, the finaltensegrity system has a repeatable, consistent shape.

A tensegrity robot according to some embodiments of the invention isshown in FIG. 1. According to some embodiments of the invention, atensegrity robot 100 includes a plurality of compressive members 102;and a plurality of interconnecting tensile members 104 connected to saidplurality of compressive members 102 to form a spatially definedstructure without said plurality of compressive members 102 formingdirect load-transmitting connections with each other. The plurality ofinterconnecting tensile members 104 form a lattice, and the latticecomprises an elastic material.

According to some embodiments, the plurality of interconnecting tensilemembers have an integral structure. FIG. 6 shows an example of aplurality of interconnecting tensile members having an integralstructure.

According to some embodiments, each of the plurality of interconnectingtensile members has a same length. For example, each side of thetriangle lattice in FIG. 5 has the same length. Each side of thetriangle lattice in FIG. 5 may be considered a tensile member.Accordingly, the embodiment shown in FIG. 5 is a lattice comprises threetensile members, while the embodiment shown in FIG. 6 comprises 24tensile members.

According to some embodiments, each of the plurality of interconnectingtensile members connects one of the plurality of compressive members toanother of the plurality of compressive members. The compressive membersmay be referred to herein as bars or rods. FIG. 7 shows an examplewherein each tensile member connects one compressive member to anothercompressive member.

According to some embodiments, each of the plurality of interconnectingtensile members has a length that is shorter than a length of each ofthe plurality of compressive members when no force is applied to theplurality of interconnecting tensile members. For example, in thetop-left panel of FIG. 7, the compressive members are longer than thetensile members when no forces, for example, stretching forces, arebeing applied to the tensile members.

According to some embodiments, the elastic material comprises siliconerubber. According to some embodiments, the plurality of interconnectingtensile members is cut from a flat sheet of the elastic material.

According to some embodiments, the tensegrity robot further comprising aplurality of junction members, wherein each of the plurality of junctionmembers is configured to rigidly connect to one of the plurality ofcompressive members. According to some embodiments, the junction membershave a shape that allows them to connect to the ends of the compressivemembers. The junction members may also have a shape that allows them toconnect to the plurality of interconnecting tensile members.

For example, FIG. 7 shows a tensegrity robot that includes a pluralityof endcaps. Each endcap connects to an end of one of the compressivemembers. However, the embodiments of the invention are not limited tothis structure. For example, the junction members may have a differentshape, or may be integrated into the compressive members, such that thecompressive members include a feature for connecting to the tensilemembers. For example, the compressive members may be fabricated to havea knob or ridge that engages the tensile members.

According to some embodiments, the plurality of interconnecting tensilemembers includes a connection structure for connecting the plurality ofinterconnecting tensile members to one of the plurality of compressivemembers or to a junction member. According to some embodiments, theconnection structure is a loop, wherein the loop is configured toencircle one of the plurality of junction members. For example, thetensile members may have loops or rings formed therebetween, as shown inFIG. 6. The compressive member or the junction member may penetrate thering, thereby becoming fixed to the tensile members. Alternatively, theconnection structure may be a knob that engages a hollow compressivemember. The connection member may be any structure that maintains afixed relationship between an end of a compressive members and acorresponding position of the lattice.

According to some embodiments, the tensegrity robot includes sixcompressive members. According to some embodiments, each of theplurality of compressive members comprises a core rigidly fixed to aplurality of rods, each of the rods extending radially from the core.According to some embodiments, the plurality of compressive members areconnected to the plurality of interconnecting tensile members such thatthe cores of the plurality of compressive members are linearly aligned.For example, in FIG. 9B, the cores of a plurality of compressive membersare linearly aligned to form a spine structure.

FIG. 2B shows a tensegrity robot according to some embodiments of theinvention. The tensegrity robot 202 includes a plurality of compressivemembers 212; a plurality of first interconnecting tensile members 206connected to the plurality of compressive members 212 to form aspatially defined structure without the plurality of compressive members212 forming direct load-transmitting connections with each other; aplurality of second tensile members 204 connected to the plurality ofcompressive members 212, each of the plurality of second tensile members204 being in parallel to one of the plurality of first interconnectingtensile members 206; a plurality of actuators 208, each attached to oneof the plurality of compressive members 212; and a controller 210 incommunication with the plurality of actuators 208. The plurality offirst interconnecting tensile members 206 forms a lattice, wherein thelattice comprises an elastic material. Each actuator of the plurality ofactuators 208 is operatively connected to a corresponding one of theplurality of second tensile members 204 so as to selectively change atension on the corresponding one of the plurality of second tensilemembers 204 in response to commands from the controller 210 to therebychange a center of mass of the tensegrity robot 202 to effect movementthereof.

According to some embodiments, at least one of the plurality ofactuators comprises a motor driven spool to wind up and release portionsof a corresponding one of the plurality of second tensile members.According to some embodiments, the controller controls the plurality ofactuators such that two of the plurality of actuators simultaneouslychange a tension on a corresponding two of the plurality of secondtensile members to thereby change the center of mass of the tensegrityrobot to effect movement thereof. According to some embodiments, thecontroller controls the plurality of actuators such that two of theplurality of actuators alternately change a tension on a correspondingtwo of the plurality of second tensile members to thereby change thecenter of mass of the tensegrity robot to effect movement thereof.

According to some embodiments, the plurality of first interconnectingtensile members have an integral structure.

According to some embodiments of the invention, a method of forming atensegrity robot, includes cutting a plurality of interconnectingtensile members from a sheet of elastic material; and connecting theplurality of interconnecting tensile members to a plurality ofcompressive members to form a spatially defined structure without theplurality of compressive members forming direct load-transmittingconnections with each other, wherein the plurality of interconnectingtensile members forms a lattice.

According to some embodiments, the method further comprises connecting aplurality of second tensile members to the plurality of compressivemembers, each of the plurality of second tensile members being inparallel with one of the plurality of interconnecting tensile members;and connecting a plurality of actuators to the plurality of compressivemembers, each one of the plurality of actuators being operativelyconnected to a corresponding one of the plurality of second tensilemembers so as to selectively change a tension on the corresponding oneof the plurality of second tensile members to thereby change a center ofmass of the tensegrity robot to effect movement thereof.

The prototyping platform according to some embodiments is much fasterthan prior approaches. Prior approaches take hours, days, or even weeks,while this approach takes just minutes to assemble a structure.

Embodiments of the invention can result in consistent prototypes. Thismeans that the tensegrity structures are symmetric (which is difficultto achieve by hand) and that multiple mostly-identical structures can bemanufactured very quickly.

The elastic lattice used in this approach has different mechanicalproperties in comparison to a traditional spring. The elastomer canexperience plastic deformation more easily, which has a currentlyunknown effect on the use of these structures under cyclical useconditions.

Some embodiments of the prototyping platform use an elastomer that iscompatible with the laser-cutting process. This means that the elasticlattice must be made of a material that does not release toxic chemicalswhen burned or ablated away. Additionally, this elastomer should betested and calibrated for its mechanical properties when in the latticestructure, in order to make good predictions of its cyclic loadingbehavior and plastic deformation limits. Additionally, we currently onlyuse thin elastomer material, cut into straight lines. Our currentembodiment uses attachment points for the lattice, with the latticesandwiched between them, held tight by a nut and bolt assembly.According to some embodiments, two washers are used as the attachmentpoints for the lattice. This makes the lattice remain in place duringmovement. However, the embodiments of the invention are not limited tothis design.

Possible variations on this platform include the use of different typesof material, different sizes and thicknesses of material, and differentpatterns of material. This could include curved members of the latticeinstead of straight lines, as well as more complicated patterns thatoptimize a 3D shape when assembled. Additionally, many different typesof attachments between the lattice and the rigid tensegrity elementscould be used, including different levels of compression, rotation, or3D movement.

There are many successful implementations of this rapid prototypingplatform, including 6-bar spherical tensegrity structures, cubic 12-bartensegrity structures, octahedral 12-bar tensegrity structures, doublesix 12-bar spherical tensegrity structures and both type I and type IIspine-like tensegrity structures. Examples are shown in the Examplessection. Additional structures can be formed using the lattice andmethods described herein.

Finally, since this lattice is designed for use with tensegrity robots,it may be designed to be retracted by a motor to adjust its length ortension in different locations, or be paired with an additional actuatorin some way to similarly change its shape.

Although a static model is used to demonstrate the basic concept of atensegrity structure according to some embodiments, the addition ofactuators is required to gain scientific insight into the tensegrityrobot's capabilities. To do so, a 6-bar tensegrity robot with sixactuators was constructed, which is referred to as the TT-4_(mini), the4th generation spherical tensegrity robot of miniature size. FIG. 2Ashows the robot 200 prior to actuation. FIG. 2B shows the fully-actuatedrobot 202. The robot 202 makes use of small components and the modularelastic lattice to allow for rapid hardware iterations and performancetesting. The robot 202 has cables 204 in parallel with the elasticlattice 206. The cables 204 are connected to actuators 208 that canadjust the length of the cables 204. A controller 210 can be incommunication with the actuators 208 to control the actuators 208 toadjust the lengths of the cables 204.

The robot moves by adjusting the lengths of its cables with respect toits elastic lattice, thereby shifting the base of the robot with respectto its center of gravity, and causing the robot to roll forward. Withthe new tensegrity prototyping platform, we were able to quicklymanufacture and assemble a passive 6-bar tensegrity structure as thebasis of an actuated tensegrity robot. Previously, it required days toassemble a new version of a tensegrity robot. With the new technique, wewere able to assemble a new functional six-bar tensegrity robot under anhour. This platform drastically increased the rate of prototypedevelopment, which allowed us to investigate new concepts quickly.

REFERENCES

-   1. BEST Lab (2015) BEST Robotics. Retrieved Sep. 16, 2015, from    http://best.berkeley.edu/best-research/best-berkeley-emergent-space-tensegrities-robotics/-   2. Kim, K., Agogino, A., Moon, D., Taneja, L., Toghyan, A.,    Dehghani, B., Agogino, A. (2014). Rapid prototyping design and    control of tensegrity soft robot for locomotion. ROBIO.-   3. Kim, K., A. K, Agogino and A. M. Agogino, “Emergent Form-Finding    for Center of Mass Control of Ball-Shaped Tensegrity Robots,”    Proceedings of ARMS (Autonomous Robots and Multirobot Systems)    workshop, Istanbul, Turkey, May 4-5.-   5. Kim, K., A. K. Agogino, A. Toghyan, D. Moon, L. Taneja and A.M.    Agogino, “Robust Learning of Tensegrity Robot Control for Locomotion    through Form-Finding,” International Conference on Intelligent    Robots and Systems (IROS 2015), Hamburg, Germany.-   5. Chen, L.-H., P. Keegan, M. Yuen, A. M. Agogino, R. K.    Kramer, A. K. Agogino and V. Sunspiral, “Soft Robots Using Compliant    Tensegrity Structures and Soft Sensors”. ICRA Workshop on Soft    Robotics,    http://icra2015.org/conference/workshop-and-tutorial-schedule-   6. Chen, L.-H., Kim, K., Tang, E., Li, K., House, R., Jung, E.,    Agogino, A.M., Agogino, A., SunSpiral, V., “Soft Spherical    Tensegrity Robot Design Using Rod-Centered Actuation and Control”,    ASME International Design Engineering Technical Conference (IDETC)    Mechanisms and Robotics Conference, 2016.-   7. Sabelhaus, A. P., Bruce, J., Caluwaerts, K., Manovi, P.,    Firoozi, R. F., Dobi, S., Agogino, A.M., SunSpiral, V., “System    Design and Locomotion of SUPERball, an Untethered Tensegrity Robot”,    IEEE International Conference on Robotics and Automation (ICRA)    2015.-   8. Sabelhaus, A. P., Ji, H., Hylton, P., Madaan, Y., Yang, C.,    Agogino, A. M., Friesen, J., SunSpiral, V., “Mechanism Design and    Simulation of the ULTRA Spine: A Tensegrity Robot”, ASME    International Design Engineering Technical Conference (IDETC) Volume    5A: 39th Mechanisms and Robotics Conference, 2015.

The following examples describe some further concepts of the inventionwith reference to particular examples. The general concepts of thecurrent invention are not limited to the particular examples.

EXAMPLES Example 1—Modular Elastic Lattice Platform for RapidPrototyping of Tensegrity Robots

Challenging environments for robot locomotion, such as those in spaceapplications, have motivated recent research into tensegrity(tension-integrity) robots [1, 2, 3, 4, 5]. These robots include rigidelements held together in a network of cables in tension. As designedfor use in space, tensegrity robots can be made as spheres that roll ona variety of terrains [6, 7, 8, 9, 10, 11], snake-like robots whichcrawl along the ground [12, 13, 14], or as part of walking four-leggedrobots [15, 16, 17]. All of these robots are designed to exploit thevarious beneficial properties of tensegrity structures: low mass,variable stiffness [1], redundancy to failure [18], among otherbenefits.

Although tensegrity robots have the potential for robust locomotion,practical prototyping of each of the above robots has presentedchallenges. Manually attaching the springs and cables of each robotintroduces human error, and takes a long time for assembly. Thesedifficulties provide the motivation for this work.

Tensegrity structures were first introduced in the mid-1960s inarchitecture and art [19, 20, 21]. The structures' passive combinationof cables-in-tension and bars-in-compression became a significant designfeature in several architectural and sculptural structures [22, 23].

In contrast, tensegrity robots change their shape by adjusting thelengths of their cables. Many different types of tensegrity robots havebeen created, including robot designs that use pneu-matic actuators [6],shape-memory alloy actuators [24], linear motors to pull on cables [25],direct actuation via servomotors [26], as well as motors attached tospools [27]. Regardless of the actuation method used, a tensegritystructure must have all tensile elements in tension to maintain a stablestructure.

The University of California's Berkeley Emergent Space Tensegrities(BEST) Laboratory has been collaborating with the National Aeronauticsand Space Administration's (NASA) Ames Research Center on usingtensegrity structures as the basis for the next generation of spaceexploration robots [15, 7, 5]. These structures have used as sphericalrobots, designed to land and roll along different terrain, and robotspines designed to help a four-legged robot walk.

In particular, a spherical tensegrity robot has the potential to be usedas both a lander and a rover since it has the ability to passivelydistribute forces across the entire structure. The tension networkprovides shock protection from the impact of landing without requiringcomplex parachute systems while also serving as a mobility platform forexploring unpredictable environments.

Five different actuated spherical tensegrity robots have been developedwithin this collaboration: the SUPERball at NASA Ames [15], the TT-1 andTT-2 robots at UC Berkeley [7, 8], the TT-3 robot at UC Berkeley [5](FIG. 4), and the TT-4_(mini) (FIGS. 2B and 3). Each of the four “TT”robots from UC Berkeley have improvements on design, actuation, andcontrol. The TT-4_(mini) contributes a major step in manufacturing andassembly for these robots.

The UCB-NASA collaboration is extending the research of sphericaltensegrity robots to 12-bar tensegrity structures, which represents thenext largest symmetric form. We are simulating and creating rapidprototypes of two geometric forms of 12-bar structures in order to learnmore about their mobility, impact, and payload characteristics. Weanticipate that the 12-bar structure will increase the capabilities oftensegrity robots for planetary surface exploration. We present hereinthe first designs of these 12-bar robots.

Finally, tensegrity spine robots have been developed to assist thewalking of four-legged (quadruped) robots over uneven terrain. TheUnderactuated Lightweight Tensegrity Robotic Assistive Spine (ULTRASpine) is a tensegrity robot with five in-dependent vertebrae that canbend and twist, emulating a backbone's motions [15]. Though simulationsand controllers have been developed for the ULTRA Spine, the developmentof hardware prototypes has been hampered by the challenges of manuallyassembling the robot, and the difficulty in creating symmetric tensionon both sides of the spine. This invention addresses both of thosechallenges, among others.

Tensegrity structures are notoriously difficult to assemble because themembers are not in balanced compression and tension until the structureis fully assembled. In the intermediary steps of assembly, forces areunevenly distributed and the structure is difficult to constrain. It iseasy to make mistakes in assembly, such as connecting the wrong tensionand compression members to one another. To illustrate the complexity ofassembly, a low-fidelity prototype of a 6-bar tensegrity structure madewith wooden dowels and springs can take as long as an hour for a team offive people to assemble.

Additionally, it is challenging to maintain symmetric tensions in theelastic members in order to create a symmetric tensegrity structure. Forexample, using a cable in series with an extension spring for theelastic member, as is done on TT-3, requires that the cables be of equallength to achieve equal tensions. This means that the system needs to becarefully manufactured or calibrated; if not, the system is susceptibleto undesired deformation and will not perform shape-shifting maneuversas expected. Other methods for the elastic members, such as using bungeecords or other elastic materials, have the same difficulties.

The inventors examined an assembled 6-bar tensegrity structure andconceptualizing how the tension members (cables in series with springs)could be deconstructed from a 3D structure to the 2D plane. The externalgeometry of a 6-bar robot is that of an icosahedron with the tensionmembers forming a portion of the vertices. The triangular faces of thesolid could thus be mapped onto a flat sheet of material. A new elasticmedium, silicone rubber, was selected to replace the traditional cablesand springs. The new configuration was first tested using a plasticsheet, which was cut to trace the tension members of an assembled 6-bartensegrity robot. The production of this low fidelity prototype made itevident that eight triangular units, such as the one in FIG. 3, wereneeded to form the 6-bar tensegrity structure.

The first elastic prototypes of the lattice for a 6-bar sphericaltensegrity were created using 0.02 in. thick, 20 A durometer siliconerubber and cut with a single-beam Universal Systems laser cutter. Thelightness of the silicone rubber caused challenges during the lasercutting process. Because it was so light, the venting system of thelaser cutter caused the rubber to lift up and flap as it was being cut,risking the correct profile of the cut. This risk was averted by puttingmasking tape on both sides of the rubber sheet, thus making the sheetheavier so it did not lift up and flap. This ensured that the properdesign could be created without impeding the cutting ability of thelaser.

After we made a number of prototypes with this silicone lattice, itbecame clear that the 0.02 in. thick, 20A durometer silicone rubber didnot have the correct material properties for the 6-bar tensegrity; thehardness and thickness of the silicone rubber did not provide enoughtension to the system, even with different width profiles.

The prototypes in the next iteration were made with 0.0625 in. thick, 60A durometer silicone rubber. By experimenting with various widths of therubber elastic lattice, the desired tension in the system was achievedusing this material. These prototypes were produced using a double-beamUniversal Systems laser cutter. The heavier silicone rubber did not facethe same manufacturing issues as the 20 A durometer silicone rubber butpresented new difficulties in the laser cutting process. Initially thelaser cutter only etched the silicone rubber instead of cutting it. Theoptimal laser cutting setting was achieved on the cutter by using onlythe top laser beam instead of both laser beams.

The elastic prototypes made with 60 A durometer silicone rubber (FIG. 5)were much stiffer than the previous versions and could withstand highertension. Thus, these prototypes better demonstrated the uniquecharacteristics of tensegrity structures.

The benefits of the modular elastic lattice address many of thechallenges of tensegrity prototyping. As the laser cut profile of thelattice can be very quickly customized, these benefits are applicable toany tensegrity structure.

First, the lattice enables rapid manufacture and assembly. Laser cuttingis simple and fast, so the lattice is quickly produced. Assembly of thestructure with the lattice is on the order of minutes, as exemplified bythe cases of the 6-bar, 12-bar, and spine tensegrity structures.Previous methods required an hour or more. Additionally, the modularityof the lattice allows experimentation with the number of lattice piecesto optimize assembly time for a given tensegrity system.

Second, the lattice gives significant control over the system'stensions. The precision and consistency of the laser cutter results inidentical elastic members, making achieving symmetric tensions in asystem much simpler. Additionally, the spring constant of the elasticmember can be changed by adjusting the profile of the laser-cut elasticmember, and thereby the system's tensions can be designed.

We conducted tension tests with the 60 A durometer rubber to estimatehow changing the width of the elastic members alters the correspondingspring constant. The laser cutter was used to produce equal lengthstrips of the lattice material of six different widths. The experimentalsetup included securing one side of the strip and pulling on theopposite side with five different forces and recording the respectivedisplacement. Nine repetitions of this process were conducted on each ofthe widths; the resulting data is seen in Table 1. This data is usedwhen designing new lattices to estimate the width that will result inthe desired spring constant and therefore the desired tension.

TABLE 1 SPRING CONSTANT TO WIDTH COMPARISON. Width Spring Constant±Error (mm) (N/m) (N/m) 6.35 986 24.52 7.94 1472 35.56 9.53 2104 55.9611.11 2364 56.50 12.70 2812 67.75 14.29 2973 68.22

In order to demonstrate the advantages of the elastic latticeprototyping method, we describe herein the use of the lattice on the6-bar, 12-bar, and spine tensegrity structures.

The elastic lattice enabled rapid prototyping of 6-bar tensegritystructures. We experimented heavily with the modularity of the elasticlattice for this structure. We found that combining the eight trianglesinto a single piece made assembly quicker and simpler. The single-piecelattice is shown in FIG. 6. This lattice structure is then used in thedemonstration assembly shown in FIG. 7.

FIG. 7 illustrates the step-by-step sequence required to assemble a6-bar tensegrity structure using this newly developed prototypingmethod. Since the main two elements of a tensegrity structure aretension and compression, we decided to use thin-walled aluminum rods asthe compression elements in our static tensegrity prototype. Endcapswere used as the connection between the modular elastic lattice and thealuminum rods. According to some embodiments, the endcaps aremanufactured by 3D printing, though other production techniques may alsobe used. A fully assembled 6-bar tensegrity structure according to someembodiments requires one of the one-piece lattices (eight connectedrubber elastic triangle lattices), twelve of the 3D printed endcaps, andsix of the aluminum rods.

The result is a tensegrity structure that can be built in a few minutesby a single person, whereas previous 6-bar structures required 1-2 hoursand two or more people. We conducted a test in which we gave tensubjects clear instructions and asked them to assemble a 6-bartensegrity structure with the elastic lattices. It took the subjects anaverage of 77 seconds with a standard deviation of 24 seconds.

Although a static model is used to demonstrate the basic concept of atensegrity structure, the addition of actuators are required to gainscientific insight into the tensegrity robot's capabilities. To do so, a6-bar tensegrity robot with six actuators was constructed, which isreferred to as the TT-4_(mini), the 4th generation spherical tensegrityrobot of miniature size (FIGS. 2B and 3). The TT-4_(mini) makes use ofsmall components and the modular elastic lattice to allow for rapidhardware iterations and performance testing.

With the new tensegrity prototyping platform, we were able to quicklymanufacture and assemble a passive 6-bar tensegrity structure as thebasis of an actuated tensegrity robot. Previously, it required days toassemble a new version of a tensegrity robot. With the new technique, wewere able to assemble a new functional six-bar tensegrity robot under ahour. This platform drastically increased the rate of prototypedevelopment, which allowed us to experiment on new concepts quickly.

In addition to 6-bar tensegrity robots, the BEST Lab is investigating12-bar tensegrity structures as a new platform for tensegrity robots forplanetary surface exploration. There are several variations of symmetric12-bar tensegrity structures. Our lab is conducting a design study oftwo 12-bar tensegrity structures to select one that will best serve thedesign objectives of the robot. These structures are the cube and theoctahedron, so named for the shapes from which the pattern of rods ofthe structures evolve.

We created structural prototypes with the goal to gain preliminarydesign insights into their deformation and impact characteristics withphysical testing. Prototyping these structures presented significantchallenges. Each of the 12-bar structures has 36 cables and a complexgeometric form. They are also higher tension systems than the 6-barstructure. These factors can make them difficult to assemble and,important to the design study, make it difficult to achieve symmetrictensions in the elastic members.

Initial attempts at creating structural prototypes without the elasticlattice were slow and necessitated the building of jigs. It took severaldays to make simple prototypes from wooden dowels and elastic bands.Regular tension and structural robustness was very difficult to achievein these early prototypes. Consequently, hand tests of deformation andimpact characteristics yielded low scientific return.

The rubber lattice prototyping method allowed us to build these twotensegrity structures much more quickly and with significantly morecontrol over the systems' tensions. Following a similar methodology aswas used for the 6-bar tensegrity structure, we created a lattice foreach of the 12-bar structures by observing geometric patterns anddesigning modular pieces. We estimated the appropriate profile of thepieces to achieve desired tensions in our system using the data fromTable 1. We then connected the pieces to create lattice shells. Next, weattached bars to the interior of each lattice shell to erect thetensegrity structures. The final structural prototypes are shown in FIG.8. The top structure is the octahedron, and the middle structure is thecube. Once the lattices are designed, assembling each structure takesbetween 5 and 10 minutes.

We have used these elastic lattices effectively in our design study, andbecause of the ease of assembly, robustness, and control over systemproperties that they allow, these prototypes will serve as the basis foractuated 12-bar robots.

Although most tensegrity research has involved rolling sphericalstructures, many other tensegrity shapes have robotic applications,including spine-like tensegrity robots. One tensegrity spine currentlyunder development is the ULTRA Spine [15], which is used as the body ofa quadruped robot. The ULTRA Spine experiences the same prototypingchallenges as the two spherical designs described above, and hasbenefitted from the elastic lattice prototyping platform in its mostrecent iteration.

As part of a walking quadruped robot, the ULTRA Spine is designed toassist the placement of the robot's feet using only lightweightmechanisms. The robot includes vertebrae that are attached to each otherusing a network of tensioned connectors, like other tensegritystructures. Current models of the robot bend the spine by shortening thelength of the horizontal connectors [15], visible in the FIG. 9B. Inaddition, the passive tensegrity network in the spine also givesbenefits related to quadruped walking, such as passive forcedistribution through the body, and adjustable stiffness betweendifferent legs.

The first prototype of the spine was manufactured and assembled usingcables and springs, as shown in FIG. 9A. The cables in the tensionnetwork were made of braided Dyneema, purchased off-the-shelf. Eachbraided cable is then tied to an extension spring, and its length isadjust such that the robot remains evenly tensioned. The springs andcables are fastened to the thin-walled aluminum rods using unique 3Dprinted endcaps with screws and washers. The specific componentsdescribed herein are listed as examples, and a person of ordinary skillin the art would recognize that the embodiments of the invention are notlimited to components comprising the materials, sizes, or springconstants listed here.

The assembly process for the cable tension network is not only timeconsuming, but is also very prone to error. Even with detailedinstructions, the process takes over three hours with at least twopeople measuring, cutting, and placing each of the thirty two cables.Different assembly jigs must be used at specific times during theassembly. During assembly, the cables are first loosely attached to eachvertebrae, then the saddle cables are hand tuned to maintain rotationalstability. After that, the horizontal cables are tightened until therobot is able to stand. However, due to the relationship between eachtensioned component, this process can be very tedious and inaccurate.When one saddle or horizontal cable is not the correct length, thevertebra are unevenly spaced, yielding an uneven distribution of weightacross the robot. These inconsistencies result in low scientific returnswhen cables are actuated during tests.

The lattice prototype can included the same five vertebrae, but thetensile network is maintained by the elastomer lattice jacket that wrapsaround the vertebrae (FIG. 9B). The rubber replaces cables and springsof the original prototype, eliminating many of the assembly andmanufacturing challenges. FIG. 10 illustrates the sequence required toassemble the spine tensegrity structure using one full lattice and fivevertebrae. The same thin-walled aluminum rods are used and a bolt andscrew act as endcaps that clamp onto the lattice and fit into the rods.A fully assembled spine tensegrity structure utilizes one lattice,twenty bolt endcaps, and twenty of the aluminum rods.

With the new prototyping platform, the total assembly time for thespinal tensegrity structure was reduced from around three hours to sevenminutes, even with a single person. A simple and easily visualizedpattern reduces the complexity of the system and allows for the assemblyprocess to be quickly learned with limited direction. The latticecreates a consistent and repeatable tension network that can be usedwhen evaluating the spine's design. After applying a force to create thebending or torsional moment, the lattice allows the robot to return toits original shape through its control of the shape or profile of therobot and the tension network established by the elastomer.

Elastic lattice designs for all three types of tensegrity robots wereassembled multiple times to quantify improvements in their use. Table 2shows the results of these trials and compares them with the generalassembly times for previous robot designs, performed by members of theBEST Lab who had experience assembling traditional cable-and-springrobots. The elastic lattices significantly reduced assembly times by asmuch as an order of magnitude for the 6-bar and spine robots.

TABLE 2 ROBOT ASSEMBLY TIME COMPARISON Robot Old Method Elastic Lattice6-Bar Sphere 1-2 hrs 77 s ± 24 s (N = 10) 12-Bar Sphere 1-2 hrs 5-10 minSpine 3-5 hrs 6 min 14 s ± 97 s (N = 8)

The newly developed rapid prototyping method using modular elasticlattices has simplified the traditional methods of building tensegritystructures. As such, we were able to shorten the time for assembly of astatic structure from one hour to within a few minutes. In addition, wewere able to modify the static structure into an actuated robot byattaching actuators and a controller; the total assembly time of anactuated robot using this prototyping platform is less than an hour. Inaddition, the examples described herein illustrate the extensibility ofthe platform for related applications, such as the rapid prototyping of12-bar and spine tensegrity structures. For researchers, this rapidprototyping platform can significantly reduce the complexity ofconstructing tensegrity structures.

REFERENCES—EXAMPLE 1

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State estimation for tensegrity robots. In 2016 IEEE    International Conference on Robotics and Automation (ICRA), pages    1860-1865. IEEE, May 2016. ISBN 978-1-4673-8026-3.    doi:10.1109/ICRA.2016.7487331. URL    http://arxiv.org/abs/1510.0124http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?armumber=7487331.-   [4] Jonathan Bruce, Andrew P Sabelhaus, Yangxin Chen, Dizhou Lu,    Kyle Morse, Sophie Milam, Ken Caluwaerts, Alice M. Agogino, and    Vytas SunSpiral. SUPER-ball: Exploring Tensegrities for Planetary    Probes. In 12th International Symposium on Artificial Intelligence,    Robotics and Automation in Space (i-SAIRAS), 2014. URL    http://robotics.estec.esa.int/i-SAIRAS/isairas2014/Data/Session5c/ISAIRAS{    }FinalPaper{ }0107.pdf.-   [5] Lee-huang Chen, Kyunam Kim, Ellande Tang, Kevin Li, Richard    House, Erik Jung, Alice M Agogino, Adrian Agogino, and Vytas    SunSpiral. Soft Spherical Tensegrity Robot Design Using Rod-Centered    Actuation and Control. 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Robust learning of tensegrity    robot control for locomotion through form-finding. In 2015 IEEE/RSJ    International Conference on Intelligent Robots and Systems (IROS),    pages 5824-5831. IEEE, September 2015. ISBN978-1-4799-9994-1. doi:    10.1109/IROS.2015.7354204. URL    http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=73    54204.-   [9] Atil Iscen, Adrian Agogino, Vytas SunSpiral, and Kagan Turner.    Flop and Roll: Learning Robust Goal-Directed Locomotion for a    Tensegrity Robot. In Proceedings of The 2014 IEEE/RSJ International    Conference on Intelligent Robots and Systems (IROS 2014), 2014.-   [10] Atil Iscen, Adrian Agogino, Vytas SunSpiral, and Ka-gan Turner.    Learning to Control Complex Tensegrity Robots. In International    Conference on Autonomous Agents & Multiagent Systems (AAMAS), pages    1193-1194, 2013. URL    http://dl.acm.org/citation.cfm?d=2484920.2485138.-   [11] Atil Iscen, Ken Caluwaerts, Jonathan Bruce, Adrian Agogino,    Vytas SunSpiral, and Kagan Turner. Learning Tensegrity Locomotion    Using Open-Loop Control Signals and Coevolutionary Algorithms.    Artificial Life, 21(2):119-140, May 2015. ISSN 1064-5462. doi:    10.1162/ARTLa 00163. URL-   [12] Brian R Tietz, Ross W Carnahan, Richard J Bachmann, Roger D    Quinn, and Vytas SunSpiral. Tetraspine: Robust terrain handling on a    tensegrity robot using central pattern generators. In 2013 IEEE/ASME    International Conference on Advanced Intelligent Mechatronics, pages    261-267. IEEE, July 2013. ISBN 978-1-4673-5320-5. doi:    10.1109/AIM.2013.6584102. URL    http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6584102.-   [13] B Mirletz, In-Won Park, Thomas E Flemons, Adrian K Agogino,    Roger D Quinn, and Vytas SunSpiral. Design and Control of Modular    Spine-Like Tensegrity Structures. 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Example 2—Inclined Surface Locomotion Strategies for SphericalTensegrity Robots

According to some embodiments of the invention, a teleoperated sphericaltensegrity robot is capable of performing locomotion on steep inclinedsurfaces. With a novel control scheme centered around the simultaneousactuation of multiple cables, the robot demonstrates robust climbing oninclined surfaces in hardware experiments and speeds significantlyfaster than previous spherical tensegrity models. This robot is animprovement over other iterations in the TT-series and the firsttensegrity to achieve reliable locomotion on inclined surfaces of up to24°. We analyze locomotion in simulation and hardware under single andmulti-cable actuation, and introduce two novel multi-cable actuationpolicies, suited for steep incline climbing and speed, respectively. Wepropose compelling justifications for the increased dynamic ability ofthe robot and motivate development of optimization algorithms able totake advantage of the robot's increased control authority.

A tensegrity structure according to some embodiments includes rodssuspended in a network of cables, where the rods and cables experienceonly compression and tension, respectively, while in equilibrium.Because there are no bending moments, tensegrity systems are inherentlyresistant to failure [1]. Additionally, the structures are naturallycompliant, exhibiting the ability to distribute external forcesthroughout the tension network. This mechanical property provides shockprotection from impact and makes the structure a robust robotic platformfor mobility in an unpredictable environment. Thus, tensegrity robotsare a promising candidate for exploration tasks, especially in the realmof space exploration, because the properties of tensegrity systems allowthese robots to fulfill both lander and rover functionality during amission.

Analysis of tensegrity robotic locomotion on inclined terrain iscritical in informing path-planning and trajectory tracking decisions inmission settings. Despite the crucial role of uphill climbing inplanetary exploration, the TT-4mini robot is the first untetheredspherical tensegrity robot to achieve reliable inclined surfaceclimbing.

The TT-4_(mini) robot was rapidly constructed using a novel modularelastic lattice tensegrity prototyping platform [2], which allows forrapid hardware iterations and experiments.

This work presents the simulation results of inclined uphill locomotionfor a six-bar spherical tensegrity robot as well as the prototyping andhardware experiments performed to validate these results. We show thatthe TT-4_(mini) robot can achieve robust locomotion on surface inclinesup to 24° using a two-cable actuation scheme in hardware, as shown inFIG. 11.

Herein, we first describe the topology and design of the TT-4_(mini),which uses a novel rapid tensegrity prototyping method. Next, we analyzeincline locomotion performance in simulation under a single-cableactuation policy. This policy is tested on hardware to establish aperformance benchmark against which two-cable actuation policies can beevaluated. Two variants of multi-cable policies are found in simulation,one suited for steep inclines and the other suited for speed. Wedemonstrate significant performance improvements in both tasks over thesingle-cable benchmark and discuss the primary factors that lead toimproved performance.

Tensegrity robots have become a recent subject of interest due to theirapplications in space exploration [3]. The natural compliance andreduced failure modes of tensegrity structures have motivated thedevelopment of multiple tensegrity robot forms [1]. Some examplesinclude spherical robots designed for locomotion on rugged terrain [4],[5], [6], snake-like robots that crawl along the ground [7], andassistive elements in walking quadrupedal robots [8], [9], [10].

Tensegrity locomotion schemes have been studied in both the context ofsingle-cable actuation [11], and (rarely) in the context of multi-cableactuation [12]. However, much of this exploration into tensegritymulti-cable actuation policies has been in the context of vibrational,rather than rolling motion.

While there exists extensive prior work in incline robotic locomotion,the literature does not directly address tensegrities. For example,Stanford's spacecraft/rover hybrid robot has demonstrated throughsimulation and hardware tests the potential for uphill locomotion.Rather than a tensegrity mechanism, however, Stanford's hybrid robotuses a flywheel-based hopping mobility mechanism designed for traversingsmall micro-gravity bodies [13].

Movement on rough or uphill terrain is a frequent occurrence in spaceexploration, and has proven to be a necessary challenge for traditionalwheeled rovers. For instance, Opportunity has ascended, with muchdifficulty, a number of surfaces up to 32° above horizontal [14]. On theother hand, NASA's SUPERball, which is also a 6-bar tensegrity robot,has demonstrated successful navigation of an 11.3° (20% grade) inclinein simulation [15]. However, as will be discussed later, the TT-4. isthe first tensegrity robot to successfully demonstrate significantinclined surface locomotion, not only in simulation, but also inhardware testing.

In order to greatly simplify and expedite the process of assembly wedeveloped a modular elastic prototyping platform for tensegrity robots[2]. The TT-4., a six-bar spherical tensegrity robot, was the firsttensegrity robot assembled using this new prototyping platform and canbe rapidly assembled in less than an hour by a single person. Toconstruct the robot, a regular icosahedron structure is first rapidlyassembled using the modular elastic lattice platform and six aluminumrods of 25 cm each, creating the passive structure of the tensegrityrobot. A total of six actuators and a central controller are thenattached to the structure, resulting in a dynamic, underactuatedtensegrity robot, as shown in FIG. 2B.

A spherical tensegrity robot can perform rudimentary punctuated rollinglocomotion by contracting and releasing each of its cables in sequence,deforming its base and shifting its center of mass (CoM) forward of thefront edge of its supporting base polygon. This contraction places therobot in a transient, unstable state, from which it naturally rolls ontothe following stable base polygon. After the roll, the robot releasesthe contracted cable and returns to its neutral stance before initiatingthe next step in the sequence. Herein, the neutral stance of the robotrefers to the stance in which no cables are contracted and the onlytension in the system is due to gravity.

While other robots have successfully achieved punctuated rolling on flatground using this technique [16], [15], we show that the TT-4_(mini) isnot only capable of the same, but can also do so on an inclined surface.

As there had been very little previous work on uphill climbing withspherical tensegrity robots, we first validated the actuation policy insimulation. Using the NASA Tensegrity Robotics Toolkit (NTRT) simulationframework, we simulated the single-cable TT-4_(mini) actuation scheme(FIG. 12) for uphill climbing on surfaces of varying inclines. Resultsshowed that the robot could successfully climb an incline of 16° insimulation using a single-cable actuation policy.

Simulation results at this incline are shown in FIG. 13. Beyond 16°, wefound that the robot could no longer reliably perform locomotion, forthe following two reasons: (1) The robot was unable to move theprojected CoM sufficiently forward to initiate an uphill roll, and (2)Deformation of the base polygon shifted the CoM behind the back(downhill) edge of the polygon, initiating a downhill roll.

To analyze the limitations of single-cable actuation policies, westudied the relationship between actuation efficiency and incline angleusing simulated sensor data. At each angle of inclination, we recordedthe cable actuation required to initiate rolling, as a fraction ofinitial cable length. As expected, we found that initiating tipping ofthe robot at greater angles of inclination requires greater cablecontraction (FIG. 14). Interestingly, the extent to which the angle ofinclination affects the required cable contraction is dependent on whichparticular cable is being actuated. Due to the inherent symmetry of the6-bar spherical structure, the TT-4_(mini)'s repeating six-step gait canbe separated into two repeated three step sub-sequences, which arisefrom the uneven, yet symmetric, distribution of tensions in the springssuspending the central payload (in this case, the central controller).

Our results imply that climbing steeper hills requires greater powerconsumption and more careful motion, motivating the development of moreefficient actuation policies for uphill locomotion. This analysishighlights the mechanical limits of single-cable actuation policies,thus encouraging exploration of alternative actuation policies.

In order to validate the results from software simulations, weconstructed an adjustable testing platform which allowed for incrementaladjustments of the surface incline angle. Using this setup, weconsidered as successful those trials in which the TT-4.i was able toreliably travel 91.4 cm (3 ft) along the inclined plane. We consideredas failure those trials in which the TT-4_(mini) failed to reach the91.4 cm mark.

We found that the robot was able to successfully perform uphill climbingup to 13° in hardware with a single-cable actuation policy. Beyond 13°,relaxing a member after its successful contraction consistently shiftedthe CoM beyond the robot's backward tipping point, causing the structureto roll down the incline. The coefficient of static friction between therobot and the surface, measured for all 8 stable robot poses, rangedfrom 0.42 to 0.57 with a mean of 0.49. This corresponds to maximuminclines before slipping ranging from 23° to 29°, with a mean of 26°. Webelieve the reason for this range is due to the lack of materialhomogeneity at contact points between the robot and the ground, whichcan include some combination of the rubber lattice and metal end-cap. Inaddition, as the distribution of weight on the end of the rods changeswith the robot's orientation, it is likely that the frictional forcesfor each face are not uniform.

Based on these results, we did not expect, nor did we observe anyfailure due to sliding in the single cable actuation tests. However, aswill be discussed in later sections, this does become a limiting factorin the robot's performance at much steeper inclines. These results areconsistent with failure modes observed in simulation.

As a baseline for comparison in later sections, the robot's averagevelocity was recorded when travelling 91.4 cm along a 10° incline.Across 10 trials under these conditions, the TT-4mini achieved anoverall average velocity of 1.96 cm/s. For reference the robot has a rodlength of 25 cm. These results serve as the first demonstration of atensegrity robot reliably climbing an inclined surface.

Having reached the limits of inclined locomotion for the single-cableactuation policy, the following actuation policies were explored.

-   -   Simultaneous actuation policy: Similar to single-cable        actuation, except the next cable contracts as the current        releases, allowing for more steps to be made in less time. See        FIG. 15, upper plot.    -   Alternating actuation policy: To preserve a low center of mass        during uphill rolling, the next cable is fully contracted before        the current is released. See FIG. 15, lower plot.

We found that multi-cable actuation policies allow the robot to climbsteeper inclines and travel at significantly faster speeds than thesingle-cable actuation policy. The following sections present theperformance results of two-cable actuation policies in simulation, andtheir validation through hardware experiments, summarized in the tablein FIG. 16.

The two-cable actuation policies, as described above, were implementedand tested in NTRT as open-loop controllers using the same robot modeland inclined surfaces as the aforementioned single-cable simulations.These simulations demonstrated vast improvements in incline locomotionstability as well as average speed, with the robot able to navigateinclines up to 26° using alternating two-cable actuation (FIG. 17) and24° using simultaneous two-cable actuation.

The significant performance improvements achieved with the two-cablepolicies are primarily due to the increased stability of the robot andits subsequent ability to avoid rolling downhill during actuation. Webelieve that this is due to a combination of two primary factors, namelyCoM height and number of contact points between the robot and theground. From the simulation results in FIG. 18 it was observed that theaverage CoM was consistently lower throughout the actuation sequence ofthe robot, especially at the critical moments approaching the tippingpoint. On a flat surface, it was found that the maximum CoM heights were93.1% and 79.8% of the neutral stance CoM height for simultaneous, andalternating actuation, respectively.

In addition to the lower CoM, both two-cable policies maintain at leastone cable in contraction at all times. In contrast to the three contactpoints in single-cable actuation, the contracted cable keeps the robotin a perpetually forward-leaning stance with four points of contact withthe ground, resulting in a larger supporting base polygon (the convexhull of the four contact points), as illustrated by FIG. 20.

Moreover, the stance of robot places the projected CoM uphill of thecentroid of the base polygon and farther away from the downhill edge, asopposed to behind it as in the single cable case. This leads to adrastic improvement in incline stability, as the robot is less likely toroll backwards due to external disturbances. Conversely, this also meansthat it is easier for the robot to roll forwards, as the distance tomove the projected CoM outside the supporting polygon in the desireddirection is smaller and therefore easier to achieve. This is especiallyapparent in FIG. 20, where the CoM is 51.4% closer to the uphill edgewhen compared to the single-cable case. The stances of single-cable andtwo-cable actuation are shown in FIG. 19.

As the robot no longer returns to a neutral state before initiating thenext roll sequence, the simultaneous policy saw a notable increase inaverage speed. However, it did not appear that the increased speed hasmuch effect on the robot's ability to navigate an incline, as thepunctuated manner in which actuation is performed means that little ifany momentum is preserved from one roll to the next.

In accordance with simulation results, the ability of the robot toactuate multiple cables simultaneously and in alternating order resultedin significant improvements in its ability to navigate steep inclinesand achieve high speeds.

The TT-4_(mini) was able to leverage alternating two-cable actuation toreliably climb a 24° (44.5% grade) incline, far outperforming therobot's previous best of 13° (23.1% grade) set via single-cableactuation. Such a significant improvement establishes this performanceas the steepest incline successfully navigated by a spherical tensegrityrobot to date. Indeed, the primary cause for failure of two-cablealternating actuation at and beyond 24° was not falling backwards, butrather slipping down the slope due to insufficient friction, inaccordance with our measurements mapping the robot's mean coefficient offriction to a theoretical max incline of 26°. This suggests that furtherimprovements may be made to the robot's incline rolling ability givencareful consideration of material choices in the next design iteration.

Not only did the robot's incline climbing performance improve, but itslocomotion speed did as well. As mentioned previously, on an incline of10°, the traditional, single-cable actuation policy traveled a distanceof 91.4 cm with an average velocity of 1.96 cm/s. However, whenperforming simultaneous two-cable actuation, the robot was able totravel the same distance with a 10-trial average velocity of 4.22 cm/s,achieving an increase of over 115% beyond the single-cable baseline. Weanticipate that this improvement can be increased by further overlappingthe contractions and relaxations of more cables in the simultaneousactuation policy. As the number of cables being simultaneously actuatedincreases, the rolling pattern increasingly resembles a fluid, sphericalroll. However, more complex actuation patterns also require anincreasingly skilled robot teleoperator. We recognized that an increasein operator skill leads to an increase in performance, but this alsoindicates the great potential for intelligent policy optimization andautomation. This has the potential to far outperform human operators andachieve ever faster locomotion and the conquering of steeper inclines.

We have demonstrated, through both simulation and hardware results, theability of a spherical tensegrity robot to perform consistent uphilllocomotion on steep inclines. This was made possible through thedevelopment of a novel multi-cable actuation scheme, which allowed theTT-4_(mini) to reliably perform forward locomotion on much steeperinclines and at greater speeds than what was previously possible usingonly single-cable actuation.

Due to the inherent coupled, nonlinear dynamics of the robot,multi-cable actuation policies render robotic control a challengingintellectual task, providing a launch point for future work. Artificialintelligence (particularly evolutionary algorithms and deepreinforcement learning architectures) may be integrated in this roboticplatform to optimize locomotive gaits on varied inclines, and evengenerate optimal tensegrity topologies, areas which have provenpromising in prior work [17], [18]. Learning algorithms may be leveragedto achieve more fluid and efficient locomotion using a robust and fullyautonomous control policy.

REFERENCES—EXAMPLE 2

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The embodiments illustrated and discussed in this specification areintended only to teach those skilled in the art how to make and use theinvention. In describing embodiments of the invention, specificterminology is employed for the sake of clarity. However, the inventionis not intended to be limited to the specific terminology so selected.The above-described embodiments of the invention may be modified orvaried, without departing from the invention, as appreciated by thoseskilled in the art in light of the above teachings. It is therefore tobe understood that, within the scope of the claims and theirequivalents, the invention may be practiced otherwise than asspecifically described.

1. A tensegrity robot, comprising: a plurality of compressive members;and a plurality of interconnecting tensile members connected to saidplurality of compressive members to form a spatially defined structurewithout said plurality of compressive members forming directload-transmitting connections with each other, wherein said plurality ofinterconnecting tensile members forms a lattice, and wherein saidlattice comprises an elastic material.
 2. The tensegrity robot accordingto claim 1, wherein said plurality of interconnecting tensile membershave an integral structure.
 3. The tensegrity robot according to claim1, wherein each of said plurality of interconnecting tensile members hasa same length.
 4. The tensegrity robot according to claim 1, whereineach of said plurality of interconnecting tensile members connects oneof said plurality of compressive members to another of said plurality ofcompressive members.
 5. The tensegrity robot according to claim 1,wherein each of said plurality of interconnecting tensile members has alength that is shorter than a length of each of said plurality ofcompressive members when no force is applied to said plurality ofinterconnecting tensile members.
 6. The tensegrity robot according toclaim 1, wherein said elastic material comprises silicone rubber.
 7. Thetensegrity robot according to claim 1, wherein said plurality ofinterconnecting tensile members is cut from a flat sheet of said elasticmaterial.
 8. The tensegrity robot according to claim 1, furthercomprising a plurality of junction members, wherein each of saidplurality of junction members is configured to rigidly connect to one ofsaid plurality of compressive members.
 9. The tensegrity robot accordingto claim 1, wherein said plurality of interconnecting tensile membersincludes a connection structure for connecting said plurality ofinterconnecting tensile members to one of said plurality of compressivemembers or to a junction member.
 10. The tensegrity robot according toclaim 9, wherein said connection structure is a loop, wherein said loopis configured to encircle one of said plurality of junction members. 11.The tensegrity robot according to claim 1, wherein said tensegrity robotincludes six compressive members.
 12. The tensegrity robot according toclaim 1, wherein each of said plurality of compressive members comprisesa core rigidly fixed to a plurality of rods, each of said rods extendingradially from said core.
 13. The tensegrity robot according to claim 12,wherein said plurality of compressive members are connected to saidplurality of interconnecting tensile members such that said cores ofsaid plurality of compressive members are linearly aligned.
 14. Atensegrity robot, comprising: a plurality of compressive members; aplurality of first interconnecting tensile members connected to saidplurality of compressive members to form a spatially defined structurewithout said plurality of compressive members forming directload-transmitting connections with each other; a plurality of secondtensile members connected to said plurality of compressive members, eachof said plurality of second tensile members being in parallel to one ofsaid plurality of first interconnecting tensile members; a plurality ofactuators, each attached to one of said plurality of compressivemembers; and a controller in communication with said plurality ofactuators, wherein said plurality of first interconnecting tensilemembers forms a lattice, wherein said lattice comprises an elasticmaterial, and wherein each actuator of said plurality of actuators isoperatively connected to a corresponding one of said plurality of secondtensile members so as to selectively change a tension on saidcorresponding one of said plurality of second tensile members inresponse to commands from said controller to thereby change a center ofmass of said tensegrity robot to effect movement thereof.
 15. Thetensegrity robot of claim 14, wherein at least one of said plurality ofactuators comprises a motor driven spool to wind up and release portionsof a corresponding one of said plurality of second tensile members. 16.The tensegrity robot of claim 15, wherein said controller controls saidplurality of actuators such that two of said plurality of actuatorssimultaneously change a tension on a corresponding two of said pluralityof second tensile members to thereby change said center of mass of saidtensegrity robot to effect movement thereof.
 17. The tensegrity robot ofone of claim 14, wherein said controller controls said plurality ofactuators such that two of said plurality of actuators alternatelychange a tension on a corresponding two of said plurality of secondtensile members to thereby change said center of mass of said tensegrityrobot to effect movement thereof.
 18. The tensegrity robot according toclaim 14, wherein said plurality of first interconnecting tensilemembers have an integral structure.
 19. A method of forming a tensegrityrobot, comprising: cutting a plurality of interconnecting tensilemembers from a sheet of elastic material; and connecting said pluralityof interconnecting tensile members to a plurality of compressive membersto form a spatially defined structure without said plurality ofcompressive members forming direct load-transmitting connections witheach other, wherein said plurality of interconnecting tensile membersforms a lattice.
 20. The method forming a tensegrity robot according toclaim 19, further comprising: connecting a plurality of second tensilemembers to said plurality of compressive members, each of said pluralityof second tensile members being in parallel with one of said pluralityof interconnecting tensile members; and connecting a plurality ofactuators to said plurality of compressive members, each one of saidplurality of actuators being operatively connected to a correspondingone of said plurality of second tensile members so as to selectivelychange a tension on said corresponding one of said plurality of secondtensile members to thereby change a center of mass of said tensegrityrobot to effect movement thereof.